经典相对论性粒子场系统的高阶场论和弱Euler-Lagrange-Barut方程
High-order Field Theory and Weak Euler-Lagrange-Barut Equation for Classical Relativistic Particle-Field Systems
摘要：
人们普遍认为，守恒定律，特别是能量-动量守恒，对物理学中的经典和量子系统都具有根本的重要性。一个广泛使用的方法来推导守恒律是基于诺特定理。然而，对于经典的相对论性粒子场系统，这一过程仍然受到阻碍。与量子情形不同的是，当我们把粒子场看作经典世界线时，障碍就出现了。困难来自两个方面。本研究发展了经典相对论性粒子场系统的一般几何（显协变）场论。在考虑质量壳约束时，将Euler-Lagrange-Barut（ELB）方程作为Euler-Lagrange（EL）方程的几何形式，用于确定相对论粒子的世界线。作为标准场论中的微分方程，对称条件的无穷小判据被转化为积分微分方程。为了克服第二个困难，我们在四维时空上建立了一个弱ELB方程。弱形式的ELB方程将在建立对称性和局部守恒律之间的联系方面发挥重要作用。使用场论与弱ELB方程在这里开发，守恒定律可以系统地从系统的对称性，承认。
英文摘要：
It is widely accepted that conservation laws, especially energy-momentum conservation, have fundamental importance for both classical and quantum systems in physics. A widely used method to derive the conservation laws is based on Noether's theorem. However, for classical relativistic particle-field systems, this process is still impeded. Different from the quantum situation, the obstruction emerged when we regard the particle's field as a classical world line. The difficulties come from two aspects. One is the mass-shell constraint and the other comes from the heterogeneous-manifolds that particles and fields reside on. This study develops a general geometric (manifestly covariant) field theory for classical relativistic particle-field systems. In considering the mass-shell constraint, the Euler-Lagrange-Barut (ELB) equation as a geometric version of the Euler-Lagrange (EL) equation is applied to determine the world lines of the relativistic particles. As a differential equation in the standard field theory, the infinitesimal criterion of the symmetry condition is converted into an integro-differential equation. To overcome the second difficulty, we develop a weak ELB equation on the 4D space-time. The weak version of the ELB equation will play an essential role in establishing the connections between symmetries and local conservation laws. Using field theory together with the weak ELB equation developed here, the conservation laws can be systematically derived from the symmetries that the systems admit.
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